Fractal layout of data blocks across multiple devices

ABSTRACT

A system, method, and computer-readable storage medium for mapping block numbers within a region to physical locations within a storage system. Block numbers are mapped within a region according to a fractal-based space-filling curve. If the region is not a 2 k  by 2 k  square, then the region is broken up into one or more 2 k  by 2 k  squares. Any remaining sub-region is centered within a 2 k  by 2 k  square, the 2 k  by 2 k  square is numbered using a fractal-based space-filling curve, and then the sub-region is renumbered by assigning numbers based on the order of the original block numbers of the sub-region.

BACKGROUND

1. Field of the Invention

This invention relates to storage systems and, more particularly, tomapping data blocks across multiple storage devices.

2. Description of the Related Art

Storage systems store and retrieve data in response to input/output(I/O) requests received from the clients. Storage systems often includephysical volumes, which may be actual physical storage devices, such ashard disks, solid-state devices, storage devices using another storagetechnology, or partitions of a storage device. Software applications,such as logical volume managers or disk array managers, provide a meansof allocating space within a storage system. Storage virtualization maybe utilized within a storage system to provide an abstraction of logicalstorage from physical storage in order to access logical storage withoutend-users identifying physical storage.

To support storage virtualization, a volume manager performs I/Oredirection by translating incoming I/O requests using logical addressesfrom end-users into new requests using addresses associated withphysical locations in the storage devices. Mapping tables may beutilized to perform the I/O redirection by mapping from block numbers ina logical address space to physical locations within the storagedevices.

Referring now to FIG. 1, four storage devices 100A-D are shown. Thesestorage devices may be part of a storage subsystem, and the storagesubsystem may be utilized by any number of clients. Each storage device100A-D includes a corresponding region 102A-D, respectively. Each regionmay be a logical address space that maps to a corresponding physicaladdress space that may be utilized for storing data.

Turning now to FIG. 2, one prior art approach for mapping virtual blocknumbers to a plurality of storage devices is shown. Virtual blocknumbers are generated for the virtual blocks of regions 102A-D ofstorage devices 100A-D of FIG. 1. It is noted that “virtual blocknumbers” may be referred to as “block numbers” and “virtual blocks” maybe referred to as “blocks” herein. In the example shown, consecutiveblocks are mapped to the same region, such that blocks 0-3 are mapped toregion 102A, blocks 4-7 are mapped to region 102B, blocks 8-11 aremapped to region 102C, and blocks 12-15 are mapped to region 102D. Thismapping scheme assigns blocks such that consecutive blocks are stored onthe same storage device. This may ensure good sequential readperformance, but limits the amount of parallelism used until the I/Orequest size is very large. For example, a request for six blocks wouldlikely involve only two storage devices, though it could involve a thirdin some cases (e.g., blocks 3-8).

Turning now to FIG. 3, another prior art approach for mapping blocknumbers to a plurality of storage devices is shown. In this approach,block numbers are generated such that consecutive blocks are stored ondifferent storage devices. I/O requests that involve multiple sequentialblocks are guaranteed to use different devices, increasing the amount ofparallel accesses to storage. However, in some cases, there may be toomuch parallelism in the resultant I/O requests. For example, any requestof four or more blocks would involve all four storage devices. As aresult, too many separate I/O requests may be generated.

I/O system designers often want the number of storage devices used foran I/O request to increase as the size of the request increases, but notas fast as would be the case using the layout shown in FIG. 3. In viewof the above, improved systems and methods for mapping block numbers tostorage devices are desired.

SUMMARY OF EMBODIMENTS

Various embodiments of a computer system and methods for mapping blocknumbers to a storage system are contemplated. Block numbers aregenerated for blocks in a region that spans multiple storage devicessuch that smaller I/O requests require access to a few storage deviceswhile larger requests require access to progressively more storagedevices. A fractal layout may be utilized to layout blocks within theregion and generate the corresponding block numbers. In one embodiment,the fractal layout may be based on a Hilbert space-filling curve. Inother embodiments, other space-filling curves (e.g., a Morton curve) maybe utilized.

In some embodiments, the region corresponding to the storage system maybe a 2^(k) by 2^(k) square region, wherein k is a positive integergreater than one. In other embodiments, the region may not be a 2^(k) by2^(k) square region. In these embodiments, the region may be broken upinto smaller 2^(k) by 2^(k) square regions. Each of the smaller 2^(k) by2^(k) square regions may be numbered using a fractal-based space-fillingcurve. Any remaining portion of the overall region that is not a 2^(k)by 2^(k) square region may be positioned within a 2^(k) by 2^(k) squareregion. In one embodiment, the remaining portion may be centered withinthe 2^(k) by 2^(k) square region. The blocks of the 2^(k) by 2^(k)square region may be numbered as usual using the fractal-basedspace-filling curve, and then the blocks within the leftover portion maybe renumbered based on the original numbering scheme.

Each block of the leftover portion may have a initial number based onthe fractal-based space-filling curve numbering of the entire 2^(k) by2^(k) square region. In one embodiment, the blocks of the leftoverportion may be renumbered based on these initial numbers. The block withthe lowest initial number within the leftover portion may be renumberedto ‘0’, the block with the next lowest initial number within theleftover portion may be renumbered to ‘1’, and so on until the highestinitial number within the leftover portion is renumbered to ‘M-1’,wherein ‘M’ is the number of blocks within the leftover portion.

These and other embodiments will become apparent upon consideration ofthe following description and accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a generalized block diagram illustrating one embodiment offour storage devices.

FIG. 2 is a generalized block diagram of one embodiment of a prior artapproach for mapping block numbers to a plurality of storage devices.

FIG. 3 is a generalized block diagram of one embodiment of another priorart approach for mapping block numbers to a plurality of storagedevices.

FIG. 4 is a generalized block diagram of one embodiment of a networkarchitecture.

FIG. 5 is a generalized block diagram illustrating one embodiment of afractal-based mapping approach.

FIG. 6 is a generalized block diagram of one embodiment of a mappingtable and corresponding storage.

FIG. 7 is a generalized block diagram of one embodiment of an 8×8 blocklayout.

FIG. 8 is a generalized block diagram of one embodiment of an algorithmfor generating a space-filling curve.

FIG. 9 illustrates one embodiment of the numbering of blocks within a4×4 square region.

FIG. 10 is a generalized flow diagram of one embodiment of a method forgenerating a fractal layout of block numbers for a region utilizing aHilbert space-filling curve algorithm.

FIG. 11 is a generalized block diagram of one embodiment of arectangular region.

FIG. 12 is a generalized block diagram illustrating one embodiment of alayout of a leftover sub-region.

FIG. 13 is a generalized block diagram illustrating one embodiment of ablock numbering scheme based on a Hilbert space-filling curve.

FIG. 14 is a generalized block diagram illustrating one embodiment of aleftmost column and a rightmost column being discarded.

FIG. 15 is a generalized block diagram illustrating one embodiment of aregion with renumbered blocks.

FIG. 16 is a generalized block diagram illustrating another embodimentof a rectangular region centered within a square region.

FIG. 17 is a generalized block diagram illustrating one embodiment of arectangular region with renumbered blocks.

FIG. 18 is a generalized flow diagram illustrating one embodiment of amethod for mapping block numbers within a region.

While the invention is susceptible to various modifications andalternative forms, specific embodiments are shown by way of example inthe drawings and are herein described in detail. It should beunderstood, however, that drawings and detailed description thereto arenot intended to limit the invention to the particular form disclosed,but on the contrary, the invention is to cover all modifications,equivalents and alternatives falling within the spirit and scope of thepresent invention as defined by the appended claims.

DETAILED DESCRIPTION

In the following description, numerous specific details are set forth toprovide a thorough understanding of the present invention. However, onehaving ordinary skill in the art should recognize that the inventionmight be practiced without these specific details. In some instances,well-known circuits, structures, signals, computer program instruction,and techniques have not been shown in detail to avoid obscuring thepresent invention. It will be appreciated that for simplicity andclarity of illustration, elements shown in the figures have notnecessarily been drawn to scale. For example, the dimensions of some ofthe elements may be exaggerated relative to other elements.

This specification includes references to “one embodiment”. Theappearance of the phrase “in one embodiment” in different contexts doesnot necessarily refer to the same embodiment. Particular features,structures, or characteristics may be combined in any suitable mannerconsistent with this disclosure. Furthermore, as used throughout thisapplication, the word “may” is used in a permissive sense (i.e., meaninghaving the potential to), rather than the mandatory sense (i.e., meaningmust). Similarly, the words “include”, “including”, and “includes” meanincluding, but not limited to.

Terminology. The following paragraphs provide definitions and/or contextfor terms found in this disclosure (including the appended claims):

“Comprising.” This term is open-ended. As used in the appended claims,this term does not foreclose additional structure or steps. Consider aclaim that recites: “A computing system comprising a data storagecontroller . . . .” Such a claim does not foreclose the computing systemfrom including additional components (e.g., a network interface, one ormore processors).

“Configured To.” Various units, circuits, or other components may bedescribed or claimed as “configured to” perform a task or tasks. In suchcontexts, “configured to” is used to connote structure by indicatingthat the units/circuits/components include structure (e.g., circuitry)that performs the task or tasks during operation. As such, theunit/circuit/component can be said to be configured to perform the taskeven when the specified unit/circuit/component is not currentlyoperational (e.g., is not on). The units/circuits/components used withthe “configured to” language include hardware—for example, circuits,memory storing program instructions executable to implement theoperation, etc. Reciting that a unit/circuit/component is “configuredto” perform one or more tasks is expressly intended not to invoke 35U.S.C. §112, sixth paragraph, for that unit/circuit/component.Additionally, “configured to” can include generic structure (e.g.,generic circuitry) that is manipulated by software and/or firmware(e.g., an FPGA or a general-purpose processor executing software) tooperate in manner that is capable of performing the task(s) at issue.“Configured to” may also include adapting a manufacturing process (e.g.,a semiconductor fabrication facility) to fabricate devices (e.g.,integrated circuits) that are adapted to implement or perform one ormore tasks.

“Based On.” As used herein, this term is used to describe one or morefactors that affect a determination. This term does not forecloseadditional factors that may affect a determination. That is, adetermination may be solely based on those factors or based, at least inpart, on those factors. Consider the phrase “determine A based on B.”While B may be a factor that affects the determination of A, such aphrase does not foreclose the determination of A from also being basedon C. In other instances, A may be determined based solely on B.

Turning now to FIG. 4, a generalized block diagram of one embodiment ofa network architecture 400 is shown. As described further below, oneembodiment of network architecture 400 includes client computer systems410 a-410 b interconnected to one another through a network 480 and todata storage arrays 420 a-420 b. Network 480 may be coupled to a secondnetwork 490 through a switch 440. Client computer system 410 c iscoupled to client computer systems 410 a-410 b and data storage arrays420 a-420 b via network 490. In addition, network 490 may be coupled tothe Internet 460 or other networks through switch 450.

It is noted that in alternative embodiments, the number and type ofclient computers and servers, switches, networks, data storage arrays,and data storage devices is not limited to those shown in FIG. 4. Atvarious times one or more clients may operate offline. In addition,during operation, individual client computer connection types may changeas users connect, disconnect, and reconnect to network architecture 400.Furthermore, while the present description generally discusses networkattached storage, the systems and methods described herein may also beapplied to directly attached storage systems and may include a hostoperating system configured to perform one or more aspects of thedescribed methods. Numerous such alternatives are possible and arecontemplated. A further description of each of the components shown inFIG. 4 is provided shortly. First, an overview of some of the featuresprovided by the data storage arrays 420 a-420 b is described.

In the network architecture 400, each of the data storage arrays 420a-420 b may be used for the sharing of data among different servers andcomputers, such as client computer systems 410 a-410 c. In addition, thedata storage arrays 420 a-420 b may be used for disk mirroring, backupand restore, archival and retrieval of archived data, and data migrationfrom one storage device to another. In an alternate embodiment, one ormore client computer systems 410 a-410 c may be linked to one anotherthrough fast local area networks (LANs) in order to form a cluster. Suchclients may share a storage resource, such as a cluster shared volumeresiding within one of data storage arrays 420 a-420 b.

Each of the data storage arrays 420 a-420 b includes a storage subsystem470 for data storage. Storage subsystem 470 may comprise a plurality ofstorage devices 476 a-476 m. Storage devices 476 a-476 m may providedata storage services to client computer systems 410 a-410 c. Each ofthe storage devices 476 a-476 m may use a particular technology andmechanism for performing data storage. The type of technology andmechanism used within each of the storage devices 476 a-476 m may atleast in part be used to determine the algorithms used for controllingand scheduling read and write operations to and from each of the storagedevices 476 a-476 m. For example, the algorithms may locate particularphysical locations corresponding to the operations. In addition, thealgorithms may perform input/output (I/O) redirection for theoperations, removal of duplicate data in the storage subsystem 470, andsupport one or more mapping tables used for address redirection anddeduplication.

The logic used in the above algorithms may be included in one or more ofa base operating system (OS) 432, a volume manager 434, within a storagesubsystem controller 474, control logic within each of the storagedevices 476 a-476 m, or otherwise. Additionally, the logic, algorithms,and control mechanisms described herein may comprise hardware and/orsoftware.

Each of the storage devices 476 a-476 m may be configured to receiveread and write requests and comprises a plurality of data storagelocations, each data storage location being addressable as rows andcolumns in an array. In one embodiment, the data storage locationswithin the storage devices 476 a-476 m may be arranged into logical,redundant storage containers or redundant arrays of independent drives(RAID) for data storage and protection.

In some embodiments, each of the storage devices 476 a-476 m may utilizetechnology for data storage that is different from a conventional harddisk drive (HDD). For example, one or more of the storage devices 476a-476 m may include or be further coupled to storage consisting ofsolid-state memory to store persistent data. In other embodiments, oneor more of the storage devices 476 a-476 m may include or be furthercoupled to storage using other technologies such as spin torque transfertechnique, magnetoresistive random access memory (MRAM) technique,shingled disks, memristors, phase change memory, or other storagetechnologies. These different storage techniques and technologies maylead to differing I/O characteristics between storage devices.

In one embodiment, the included solid-state memory comprises solid-statedrive (SSD) technology. A Solid-State Disk (SSD) may also be referred toas a Solid-State Drive. Without moving parts or mechanical delays, anSSD may have a lower read access time and latency than a HDD. However,the write performance of SSDs is generally slower than the readperformance and may be significantly impacted by the availability offree, programmable blocks within the SSD.

Storage array efficiency may be improved by creating a storagevirtualization layer between user storage and physical locations withinstorage devices 476 a-476 m. In one embodiment, a virtual layer of avolume manager is placed in a device-driver stack of an operating system(OS), rather than within storage devices or in a network. Many storagearrays perform storage virtualization at a coarse-grained level to allowstoring of virtual-to-physical mapping tables entirely in memory.However, such storage arrays may not be able to integrate features suchas data compression, deduplication and copy-on-modify operations. Manyfile systems support fine-grained virtual-to-physical mapping tables,but they do not support large storage arrays, such as device groups 473a-473 m. Rather, a volume manager or a disk array manager may be used tosupport device groups 473 a-473 m.

In one embodiment, one or more mapping tables may be stored in thestorage devices 476 a-476 m, rather than memory, such as RAM 472, memorymedium 430 or a cache within processor 422. The storage devices 476a-476 m may be SSDs utilizing Flash memory. The low read access andlatency times for SSDs may allow a small number of dependent readoperations to occur while servicing a storage access request from aclient computer. The dependent read operations may be used to access oneor more indexes, one or more mapping tables, and user data during theservicing of the storage access request.

In one example, I/O redirection may be performed by the dependent readoperations. In another example, inline deduplication may be performed bythe dependent read operations. In yet another example, bulk array tasks,such as a large copy, move, or zeroing operation, may be performedentirely within a mapping table rather than accessing storage locationsholding user data. Such a direct map manipulation may greatly reduce I/Otraffic and data movement within the storage devices 476 a-476 m. Thecombined time for both servicing the storage access request andperforming the dependent read operations from SSDs may be less thanservicing a storage access request from a spinning HDD.

In addition, the information within a mapping table may be compressed. Aparticular compression algorithm may be chosen to allow identificationof individual components, such as a key within a record among multiplerecords. Therefore, a search for a given key among multiple compressedrecords may occur. In various embodiments the search for a given key maybe performed without decompressing each tuple by comparing thecompressed representation of the key against the compressed informationstored in the relevant fields of the tuple. If a match is found, onlythe matching record may be decompressed. Compressing the tuples withinrecords of a mapping table may further enable fine-grained levelmapping. This fine-grained level mapping may allow direct mapmanipulation as an alternative to common bulk array tasks. Furtherdetails concerning efficient storage virtualization will be discussedbelow.

Again, as shown, network architecture 400 includes client computersystems 410 a-410 c interconnected through networks 480 and 490 to oneanother and to data storage arrays 420 a-420 b. Networks 480 and 490 mayinclude a variety of techniques including wireless connection, directlocal area network (LAN) connections, wide area network (WAN)connections such as the Internet, a router, storage area network,Ethernet, and others. Networks 480 and 490 may comprise one or more LANsthat may also be wireless. Networks 480 and 490 may further includeremote direct memory access (RDMA) hardware and/or software,transmission control protocol/internet protocol (TCP/IP) hardware and/orsoftware, router, repeaters, switches, grids, and/or others. Protocolssuch as Fibre Channel, Fibre Channel over Ethernet (FCoE), iSCSI, and soforth may be used in networks 480 and 490. Switch 440 may utilize aprotocol associated with both networks 480 and 490. The network 490 mayinterface with a set of communications protocols used for the Internet460 such as the Transmission Control Protocol (TCP) and the InternetProtocol (IP), or TCP/IP. Switch 450 may be a TCP/IP switch.

Client computer systems 410 a-410 c are representative of any number ofstationary or mobile computers such as desktop personal computers (PCs),servers, server farms, workstations, laptops, handheld computers,servers, personal digital assistants (PDAs), smart phones, and so forth.Generally speaking, client computer systems 410 a-410 c include one ormore processors comprising one or more processor cores. Each processorcore includes circuitry for executing instructions according to apredefined general-purpose instruction set. For example, the x86instruction set architecture may be selected. Alternatively, the Alpha®,PowerPC®, SPARC®, or any other general-purpose instruction setarchitecture may be selected. The processor cores may access cachememory subsystems for data and computer program instructions. The cachesubsystems may be coupled to a memory hierarchy comprising random accessmemory (RAM) and a storage device.

Each processor core and memory hierarchy within a client computer systemmay be connected to a network interface. In addition to hardwarecomponents, each of the client computer systems 410 a-410 c may includea base operating system (OS) stored within the memory hierarchy. Thebase OS may be representative of any of a variety of operating systems,such as, for example, MS-DOS®, MS-WINDOWS®, OS/2®, UNIX®, Linux®,Solaris®, AIX®, DART, or otherwise. As such, the base OS may be operableto provide various services to the end-user and provide a softwareframework operable to support the execution of various programs.Additionally, each of the client computer systems 410 a-410 c mayinclude a hypervisor used to support virtual machines (VMs). As is wellknown to those skilled in the art, virtualization may be used indesktops and servers to fully or partially decouple software, such as anOS, from a system's hardware. Virtualization may provide an end-userwith an illusion of multiple OSes running on a same machine each havingits own resources and access to logical storage entities (e.g., LUNs)built upon the storage devices 476 a-476 m within each of the datastorage arrays 420 a-420 b.

Each of the data storage arrays 420 a-420 b may be used for the sharingof data among different servers, such as the client computer systems 410a-410 c. Each of the data storage arrays 420 a-420 b includes a storagesubsystem 470 for data storage. Storage subsystem 470 may comprise aplurality of storage devices 476 a-476 m. Each of these storage devices476 a-476 m may be an SSD. A controller 474 may comprise logic forhandling received read/write requests. A random-access memory (RAM) 472may be used to batch operations, such as received write requests. Invarious embodiments, when batching write operations (or otheroperations) non-volatile storage (e.g., NVRAM) may be used.

The base OS 432, the volume manager 434 (or disk array manager 434), anyOS drivers (not shown) and other software stored in memory medium 430may provide functionality providing access to files and the managementof these functionalities. The base OS 432 may be a storage operatingsystem such as NetApp Data ONTAP® or otherwise. The base OS 432 and theOS drivers may comprise program instructions stored on the memory medium430 and executable by processor 422 to perform one or more memory accessoperations in storage subsystem 470 that correspond to receivedrequests. The system shown in FIG. 4 may generally include one or morefile servers and/or block servers.

Each of the data storage arrays 420 a-420 b may use a network interface424 to connect to network 480. Similar to client computer systems 410a-410 c, in one embodiment, the functionality of network interface 424may be included on a network adapter card. The functionality of networkinterface 424 may be implemented using both hardware and software. Botha random-access memory (RAM) and a read-only memory (ROM) may beincluded on a network card implementation of network interface 424. Oneor more application specific integrated circuits (ASICs) may be used toprovide the functionality of network interface 424.

In addition to the above, each of the storage controllers 474 within thedata storage arrays 420 a-420 b may support storage array functions suchas snapshots, replication and high availability. In addition, each ofthe storage controllers 474 may support a virtual machine environmentthat comprises a plurality of volumes with each volume including aplurality of snapshots. In one example, a storage controller 474 maysupport hundreds of thousands of volumes, wherein each volume includesthousands of snapshots. In one embodiment, a volume may be mapped infixed-size sectors, such as a 4-kilobyte (KB) page within storagedevices 476 a-476 m. In another embodiment, a volume may be mapped invariable-size sectors such as for write requests. A volume ID, asnapshot ID, and a sector number may be used to identify a given volume.

One or more mapping tables may be used to map I/O requests from each ofthe client computer systems 410 a-410 c to physical locations in storagedevices 476 a-476 m. A first mapping table may include a mapping of avirtual address space of a given client to a logical address spacerepresented by block numbers. A second table may map the block numbersto the physical address space of storage devices 476 a-476 m. In someembodiments, the information in the first and second mapping tables maybe combined into a single mapping table. A “physical” pointer value maybe read from the second mapping table during a lookup operationassociated with a given virtual address and a corresponding blocknumber. This physical pointer value may then be used to locate aphysical location within the storage devices 476 a-476 m. It is notedthe physical pointer value may be used to access another mapping tablewithin a given storage device of the storage devices 476 a-476 m.Consequently, one or more levels of indirection may exist between thephysical pointer value and a target storage location.

In another embodiment, the mapping table(s) may comprise informationused to deduplicate data, such as deduplication table relatedinformation. The information stored in the deduplication table mayinclude mappings between one or more calculated hash values for a givendata component and a physical pointer to a physical location in one ofthe storage devices 476 a-476 m holding the given data component. Inaddition, a length of the given data component and status informationfor a corresponding entry may be stored in the deduplication table.

Referring now to FIG. 5, a block diagram of one embodiment of afractal-based layout of blocks is shown. As used herein, the term“fractal” may be defined as an object in which the parts or individualcomponents of the whole are self-similar. In other words, a “fractal” isa self-similar geometric shape produced by an equation that undergoesrepeated iterative steps or recursion. The block numbering shown in FIG.5 is generated using a fractal-based pattern to number the blocks withinthe regions 102A-D of the storage devices. In some embodiments, theregions 102A-D may be referred to as a single region.

As shown in FIG. 5, the numbering of blocks is based on a Hilbertspace-filling curve. Generally speaking, a “space-filling curve” is acurve whose range contains the entire two-dimensional addess spaceregion. In other words, a “space-filling curve” may be defined as afunction that maps a two-dimensional space into a one-dimensional space.A space-filling curve passes through every block in the two-dimensionalgrid of regions 102A-D so that each block is only visited once. Thereare many different types of space-filling curves, one of which is theHilbert space-filling curve.

The Hilbert space-filling curve is a continuous fractal space-fillingcurve first described by German mathematician David Hilbert in 1891. TheHilbert space-filling curve defines an algorithm which may be utilizedfor mapping a particular block number to a location within the region.One advantage of the Hilbert space-filling curve is that it does afairly adequate job of preserving locality. The Hilbert space-fillingcurve may take a path that traces through all the points in atwo-dimensional square grid in such a way that each step in the pathmoves between neighbors in the grid.

In other embodiments, other space-filling curves may be utilized. Forexample, in another embodiment, a Morton space-filling curve may beutilized to generate block numbers for a storage system. Depending onthe embodiment, the space-filling curve providing the best clusteringwith respect to the specific storage system may be utilized. The blocks,which correspond to a logical address space, may be numberedsequentially starting at the beginning of the space-filling curve andincrementing at each step of the curve until reaching the end of thecurve within the overall region.

Blocks may be labeled using the Hilbert space-filling curve such thatconsecutive blocks are laid out across multiple storage devices in afractal pattern. The mapping shown in FIG. 5 allows for the number ofstorage devices used for an I/O request to increase as the size of theI/O request increases, but not as fast as would be the case using thelayout shown in FIG. 3. Requests of three consecutive blocks requireaccesses to at a maximum of two different devices, while requests offour consecutive blocks usually require accesses of two differentdevices, though occasionally they may require a third device (e.g.,blocks 5-8). Requests of five blocks require accesses of at most threedifferent drives. Generally speaking, the number of storage devicesaccessed by an I/O request does not scale linearly as the number ofblocks per I/O request increases. Instead, the number of storage devicesaccessed by an I/O request of ‘N’ blocks will on average be the squareroot of N, as opposed to ‘N’ storage devices for a layout based on theparallel approach of FIG. 3.

In one embodiment, the layout shown in FIG. 5 may be generated during aconfiguration phase of the storage subsystem. This layout may be storedin a mapping table or other file. In another embodiment, the layoutshown in FIG. 5 may be generated in real-time as blocks are utilized toprocess I/O requests to the storage subsystem. In such an embodiment, aparticular block number may be mapped to a location on-the-fly based ona fractal pattern, such as the Hilbert space-filling curve. The mostrecently used block number that was utilized by a previous I/O requestmay be stored in a table or file. When a new I/O request is received,the algorithm for the fractal-based space-filling curve may start fromthis most recently used block number and generate new block numbers forthe number of blocks that are required for the new I/O request.

In a further embodiment, instead of using the next unused block number,the algorithm may jump to a further point on the fractal-basedspace-filling curve. In such an embodiment, to perform a jump to afurther point on the curve, a formula may be utilized for calculatingthe (x, y) location of a destination block in the region given thelocation along the one dimensional path (i.e. index). Formulas forcalculating these locations are well known to those skilled in the art.For example, if the last used block number is two, and a new I/O requestis received by the storage subsystem, then block numbers may begenerated starting at a block number higher than three. For example, newblock numbers may be generated starting with a block number of six. Inthis way, the algorithm may jump to a future point on the space-fillingcurve, skipping over one or more blocks that are in the path of thespace-filling curve. This is for illustrative purposes only, and anyother subsequent block number may be utilized for the I/O request. Atsome future point, the block numbers that were skipped over during thisrequest may be utilized by an I/O request.

Turning now to FIG. 6, a block diagram of one embodiment of a mappingtable 625 and corresponding storage devices is shown. Table 625 showsmappings of virtual addresses (VA) to curve block numbers (CBN) andlogical addresses (LA). The blocks of devices 605-620 may be numberedfrom 0 to 15 using a Hilbert space-filling curve as described above inregard to FIG. 5. In addition, each block includes a logical addressassociated with the location of the block within its respective storagedevice. For example, block ‘0’ is located at address ‘02’ within device605, block ‘1’ is located at address ‘09’ within device 610, and so on.The addresses shown in FIG. 6 are for illustrative purposes only, and inother embodiments, other types and representations of addresses may beutilized. For example, in another embodiment, the logical address shownin mapping table 625 may only contain a pointer to a given device, andthe device may determine the address for a specific curve block numberaccording to its own mapping mechanisms and tables.

Mapping table 625 may be stored in any of various locations (e.g.,cache, RAM, memory medium) within a storage subsystem, such as storagesubsystem 470 (of FIG. 4). Mapping table 625 may be utilized to performaddress translations from a virtual address space to the physicaladdress space of storage devices 605-620. For example, if an I/O requestwere received for virtual address (VA) 0x010002000, then this virtualaddress may correspond to curve block number (CBN) 0, which is mapped toaddress ‘02’ of storage device 605. In one embodiment, the entries andaddresses of mapping table 625 may be generated during a configurationphase of the storage subsystem. In another embodiment, the entries andaddresses of mapping table 625 may be generated in real-time. In otherembodiments, some of the entries and addresses may be generated on setupand some of the entries and addresses may be generated in real-time.

It is noted that the mapping table 625 shown in FIG. 6 is only onepossible embodiment of a mapping table which may be used to performaddress translations from virtual to the logical address space ofstorage devices 605-620. In other embodiments, mapping table 625 may beorganized in a different manner with other numbers of columns and withother types of information. Furthermore, the storage devices 605-620 mayalso have internal mapping mechanisms. For example, the logical addressshown in mapping table 625 may be mapped by the storage device to aphysical location within the device. Over time, this internal mappingbetween logical address and physical location may change.

Turning now to FIG. 7, one embodiment of an 8×8 region is shown. Theregion shown in FIG. 7 is an 8×8 square, and the columns in this layoutrepresent storage devices, or sections within storage devices, and therows represent blocks within each storage device (or section). Theregion may also be referred to as a two-dimensional grid, wherein thefirst dimension is a number of blocks and the second dimension is thenumber of storage devices. In the embodiment shown in FIG. 4, there areeight storage devices (or regions) and eight blocks per storage device.A Hilbert space-filling curve was utilized to generate the layout of thesquare in FIG. 4. In other embodiments, other space-filling curves(e.g., Morton) may be utilized. In some embodiments, the columns in theregion may actually correspond to a portion of a storage device.

The Hilbert space-filling curve may be utilized with any 2^(k) by 2^(k)square region, wherein k is a positive integer greater than one. Forexample, 16×16 squares, 32×32 squares, 64×64 squares, and so on, mayutilize a block numbering scheme based on the Hilbert space-fillingcurve. In other embodiments, other types of space-filling curves (e.g.,Morton) may be utilized to generate block numbers for the blocks of theregion.

Referring now to FIG. 8, one example of an algorithm for generating aspace-filling curve is shown. In the example shown, the code in block805 may perform a mapping from a two-dimensional grid to aone-dimensional curve. The code in block 810 may perform a mapping froma one-dimensional curve to a two-dimensional grid. The code assumes thetwo-dimensional grid is a square divided into n-by-n cells, for an ‘n’equal to a power of two, the square uses integer coordinates, with (0,0)in the lower-left corner, with (n-1, n-1) in the upper-right corner, anda distance ‘d’ that starts at 0 in the lower-left corner and goes to(n²-1) in the lower-right corner. The rotate function, which is utilizedby the code in blocks 805 and 810, is shown in block 815. However, otherways of expressing a space-filling curve algorithm are possible and arecontemplated. In one embodiment, the code shown in blocks 805-815 may beutilized for striping data across storage devices.

Referring now to FIG. 9, a block diagram of the numbering of blockswithin a 4×4 square region is shown. The 4×4 square region in FIG. 9 isnumbered based on the Hilbert space-filling curve algorithm and region825 (of FIG. 8). The path taken by the curve begins in the bottom rightcorner of the 4×4 square region, and so this block is numbered ‘0’. Theblock number is incremented for each subsequent block that is traversedby the path, such that the block above the ‘0’ block is numbered ‘1’,the block to the left of the ‘1’ block is numbered ‘2’, and so on. Theblock in the top right corner of the 4×4 square region, which is thelast block traversed by the Hilbert space-filling curve of FIG. 8, isnumbered ‘15’. The 4×4 square region of FIG. 9 may be flipped in thehorizontal direction and rotated clockwise by 90 degrees to match theblock numbering layout of the 4×4 square region shown in FIG. 5. Invarious embodiments, the block numbering layout generated by a Hilbertspace-filling curve may be flipped in various directions and/or rotatedby multiples of 90 degrees.

Turning now to FIG. 10, one embodiment of a method for generating alayout of block numbers for a region utilizing a space-filling curvealgorithm is shown. In one embodiment, a data storage controller, suchas storage controller 474 (of FIG. 4), may be configured to operate inaccordance with method 1000. For purposes of discussion, the steps inthis embodiment are shown in sequential order. It should be noted thatin various embodiments of the method described below, one or more of theelements described may be performed concurrently, in a different orderthan shown, or may be omitted entirely. Other additional elements mayalso be performed as desired.

In the embodiment shown, method 1000 begins by determining the size ofthe region which is used to for storage in a storage subsystem (block1005). For the purposes of this discussion, it will be assumed that theregion is a 2^(k) by 2^(k) square region, wherein k is a positiveinteger greater than one. For other types of regions, such as non-squareregions, other methods may be utilized which will be discussed furtherbelow. After block 1005, an algorithm may be utilized to generate aspace-filling curve (e.g., a Hilbert curve) for traversing the pluralityof blocks of the region (block 1010). In one embodiment, the algorithmshown in FIG. 8 may be utilized to generate the Hilbert space-fillingcurve. In one embodiment, the algorithm may be executed by a processor(e.g., processor 422) within a data storage array (e.g., data storagearray 420 b). In other embodiments, the algorithm may be executed by anyof various other types of hardware and/or software. Next, the blocks maybe numbered based on the path taken by the Hilbert space-filling curvethrough the region (block 1015).

After block 1015, the block numbers may be stored in a mapping table(block 1020). In one embodiment, the mapping table may be stored in astorage subsystem, such as storage subsystem 470 (of FIG. 4) of datastorage array 420 b. The mapping table may be utilized for fulfillingI/O requests received from a client. For example, a client may generatean I/O request to store a file in array 420 b. If the size of the fileis such that it corresponds to four storage blocks, the first fourblocks of the region may be utilized (0-3) for storing the fourportions, and the mapping of the blocks to the specific storage devices476 a-m may be retrieved from the mapping table to determine where tostore the four portions of the file. Subsequent files may be stored inthe remaining blocks, starting with block 4, which is the next unusedblock. This example is for illustrative purposes only, and other filesmay be broken up into any number of portions, depending on the file sizeand the size of the portions and blocks. In one embodiment, theblock-size may be the same for all blocks within the storage subsystem.In other embodiments, the block-size may vary within the storagesubsystem and/or within a single storage device.

Referring now to FIG. 11, a block diagram of a rectangular region isshown. The region shown in FIG. 11 is a rectangular-shaped 11×16 region.The 11 columns of the region correspond to 11 storage devices in thephysical address space. The 16 rows correspond to the 16 blocks that maybe stored in each storage device. The region 1100 is not a square regionwith a side length of 2^(k), and therefore one or more alternate schemesfor numbering the blocks may be utilized. In one embodiment, the overallregion 1100 may be split up into smaller regions that form squares witha side length of 2^(k). It is noted that in some embodiments, the valueof ‘k’ may vary for the smaller regions that are created by splitting upthe overall region 1100. As shown in FIG. 11, the overall region 1100may be partitioned into multiple regions, including the region 1205,which is a square of length eight (2^(k), with k=3). The overall region1100 may also be partitioned into regions 1110 and 1115, and regions1110 and 1115 may be combined to form a square of length eight.

Generally speaking, any overall region that is not a 2^(k) by 2^(k)square may be broken up into one or more smaller sized sub-regions. Theoverall region may be divided into two or more sub-regions, with atleast one of the sub-regions chosen such that it is the largest 2^(k) by2^(k) square that fits in the overall region. Then, other sub-regionsfrom the overall region may be combined, if possible, to make one ormore other 2^(k) by 2^(k) squares. These 2^(k) by 2^(k) squaresub-regions, either original 2^(k) by 2^(k) square sub-regions or acombination of two or more sub-regions that forms a 2^(k) by 2^(k)square sub-region, may have their blocks numbered according to afractal-based space-filling curve. Any leftover sub-region that cannotbe combined to form a 2^(k) by 2^(k) square sub-region may be numberedaccording to various methods depending on the embodiment, one of whichis described in further detail below.

The techniques described above may be used for regions of various sizes.For example, in another embodiment, a region may span 16 drives, and 64blocks may be stored on each drive, forming a 64×16 region. The 64×16region may be broken up into four 16×16 regions, and then each 16×16region may be mapped using a fractal-based space-filling curve. In otherembodiments, other sizes of regions may be partitioned. The techniqueused for partitioning involves first breaking up the region into thelargest square of 2^(k) side that fits into the region. If more than oneof these squares fits into the overall region, then the region may besplit up into as many squares that fit.

Also, if two or more pieces of the region may be combined to form asquare, then the region may be partitioned into these two or morepieces. For some regions, there may be a non-square shaped leftoverregion that is unable to be combined with any other piece to form asquare. For this leftover non-square sub-region, the sub-region may becentered within a 2^(k) by 2^(k) square, and then the square may benumbered using the fractal-based space-filling curve. In variousembodiments, the 2^(k) by 2^(k) square may be a different size than thepreviously used 2^(k) by 2^(k) squares. After the square has beennumbered, the rectangular leftover region, which fills only a portion ofthe square, may be renumbered. First, the leftover region may be“removed” from the overall square and then renumbered. The renumberingis accomplished via the following steps: First, look for the smallestnumber in the leftover region. This smallest number will be renumberedas ‘0’. It is possible this smallest number was already ‘0’. Then, thenext smallest number in the leftover region will be identified, and thisnext smallest number will be renumbered as ‘1’. Again, it is possiblethe number of this block was already ‘1’. These steps may continue,wherein the next higher original number is identified, and then thisblock may be renumbered with the next sequential integer in therenumbering scheme. For example, if the leftover region contains 20blocks, then these blocks may be renumbered from 0 to 19 using the stepsdescribed above. Prior to the renumbering scheme, if this leftoverregion were in an overall 8×8 square (containing 64 blocks), then theblocks may have been numbered from anywhere between 0 to 63.

Turning now to FIG. 12, a block diagram of one embodiment of a layout ofa leftover sub-region is shown. Sub-region 1120 is the leftoverrectangular region from overall region 1100 of FIG. 11. To generateblock numbers for the blocks of sub-region 1120, sub-region 1120 may becentered within 8×8 square region 1205. Then, region 1205 may benumbered using a Hilbert space-filling curve, as is shown in FIG. 13. Asis shown in FIG. 13, region 1205 is numbered in an identical fashion tothe 8×8 region shown in FIG. 6. In other embodiments, other types ofspace-filling curves may be utilized to generate a fractal layout ofblock numbers.

After region 1205 is mapped using the Hilbert space-filling curve, theleftmost and rightmost columns of region 1205 may be discarded, as isshown in FIG. 14. The leftmost and rightmost columns of region 1205include an ‘X’ in the bottom of each of the squares indicating thatthese squares are not mapped to blocks on devices. Then, the blocks ofregion 1120 may be renumbered according to the scheme previouslydescribed. The numbers in the top of each block are the original numbersused in the mapping of the entire region 1205. The numbers in the bottomof each block are the final block numbers for the blocks of region 1120,as is shown in FIG. 15. The block numbers shown in FIG. 15 may beutilized for mapping blocks to the physical address space of theunderlying storage devices. In one embodiment, the block numbers shownin FIG. 15 may be generated prior to performing the actual mapping ofblocks to the storage device and stored for later use rather than beinggenerated on-the-fly on an as-needed basis.

The techniques disclosed in FIGS. 11-15 for laying out block numbers fora rectangular region may be applied to other sizes of rectangularregions. In general, a rectangle may be of size A by B with M totalblocks, wherein A, B, and M are integers. The rectangle may bepartitioned into one or more square regions, wherein each square regionhas a side length equal to 2^(k), wherein k is a positive integergreater than one. If the rectangle is partitioned into more than onesquare region, the value of k may vary for the different square regions.The block numbers for the square region(s) may be numbered by utilizinga space-filling curve (e.g., Hilbert space-filling curve). Any leftoverregion from the original rectangle may be positioned within a square ofside length equal to 2^(k) and the square may be numbered according to aspace-filling curve. Then the rectangle portion of the square,corresponding to the leftover region, may be renumbered according to anascending sequential order of the block numbers generated for thesquare.

Turning now to FIG. 16, another embodiment of a rectangular regioncentered within a square region is shown. Region 1100 is shown in FIG.16, which is the same 11×16 region shown in FIG. 11. An alternate schemefor laying out block numbers in a fractal pattern for a rectangularregion is shown in FIG. 16. In one embodiment, region 1100 may becentered within the 16×16 square 1605. In other embodiments, region 1100may be placed at any location within square 1605. There are two extracolumns on the leftside of region 1100 and three extra columns on therightside of region 1100 which may be attached to region 1100 to make upthe overall 16×16 square 1605. In other embodiments, a region may havemore columns than rows (i.e., more storage devices than blocks). In suchan embodiment, the region may be centered within a 2^(k) by 2^(k) squaresuch that there are empty rows on top and/or on bottom of the regionwithin the square. The example shown in FIG. 16 is illustrative only,and various other sizes of regions may be centered within a 2^(k) by2^(k) square in a similar manner

In one embodiment, block numbers may be generated for region 1605 usingthe Hilbert space-filling curve. Other types of space-filling curves maybe utilized in other embodiments. Then, the extra two columns on theleft-side and the extra three columns on the right-side of region 1605may be discarded, and region 1100 may be renumbered using the schemeillustrated in FIGS. 14 and 15. The renumbering scheme for region 1100is shown in FIG. 17.

Turning now to FIG. 18, one embodiment of a method for mapping blocknumbers within a region is shown. In one embodiment, a data storagecontroller, such as storage controller 474 (of FIG. 4), may generallyoperate in accordance with method 1800. For purposes of discussion, thesteps in this embodiment are shown in sequential order. It should benoted that in various embodiments of the method described below, one ormore of the elements described may be performed concurrently, in adifferent order than shown, or may be omitted entirely. Other additionalelements may also be performed as desired.

The method 1800 may begin by determining the size of the region which isused to represent the physical address space of a storage subsystem(block 1805). The region may include any number of storage devices andany number of blocks per storage device. In one embodiment, the regionmay be represented by a two-dimensional grid, with a first dimensioncorresponding to the number of blocks and a second dimensioncorresponding to the number of storage devices. Generally speaking, I/Orequests may access the storage subsystem on a block-by-block basis, andso therefore, two or more storage devices of the storage subsystem maybe partitioned into blocks which may be used to process the I/Orequests.

If the region is a 2^(k) by 2^(k) square (conditional block 1810), thenthe block numbers within the region may be mapped utilizing afractal-based space-filling curve (block 1815). In one embodiment, thefractal-based space-filling curve may be a Hilbert space-filling curve.In other embodiments, other space-filling curves may be utilized.

If the region is not a 2^(k) by 2^(k) square (conditional block 1810),then the region may be positioned within a 2^(k) by 2^(k) square (block1820). The size of the 2^(k) by 2^(k) square may be chosen such that thesquare is large enough to accommodate the region. In one embodiment, theregion may be centered within the 2^(k) by 2^(k) square. In otherembodiments, the region may be placed at any location within the 2^(k)by 2^(k) square. Next, numbers may be generated for the blocks of the2^(k) by 2^(k) square utilizing a fractal-based space-filling curve(block 1825). Then, the blocks outside of the centered region may bediscarded (block 1830). In other words, the blocks that fall outside ofthe original region may be removed from consideration during thesubsequent renumbering process. Next, the blocks of the centered regionmay be renumbered from 0 to M-1, wherein M is the number of blocks inthe centered region (block 1835). The renumbering may be based on theoriginal numbering of the 2^(k) by 2^(k) square. The block numbers maybe stored in one or more mapping tables for use in processing I/Orequests directed to the storage subsystem (block 1840).

In various embodiments, one or more mapping tables may be used for I/Oredirection or translation, deduplication of duplicate copies of userdata, volume snapshot mappings, and so forth within a storage subsystem.In one embodiment, the mapping tables may be stored in the storagedevices 476 a-476 m (of FIG. 4). Furthermore, copies of portions or allof a given mapping table may be stored in RAM 472, in buffers withincontroller 474, in memory medium 430, and in one or more caches withinor coupled to processor 422.

The mapping tables may also include data fields including data such as apointer used to identify or locate data components stored in storagesubsystem 470. It is noted that in various embodiments, the storagesubsystem may include storage devices (e.g., SSDs) which have internalmapping mechanisms. In such embodiments, the pointer may not be anactual physical address per se. Rather, the pointer may be a logicaladdress which the storage device maps to a physical location within thedevice. Over time, this internal mapping between logical address andphysical location may change.

It is noted that the above-described embodiments may comprise software.In such an embodiment, the program instructions that implement themethods and/or mechanisms may be conveyed or stored on a non-transitorycomputer readable medium. Numerous types of media which are configuredto store program instructions are available and include hard disks,floppy disks, CD-ROM, DVD, flash memory, Programmable ROMs (PROM),random access memory (RAM), and various other forms of volatile ornon-volatile storage.

In various embodiments, one or more portions of the methods andmechanisms described herein may form part of a cloud-computingenvironment. In such embodiments, resources may be provided over theInternet as services according to one or more various models. Suchmodels may include Infrastructure as a Service (IaaS), Platform as aService (PaaS), and Software as a Service (SaaS). In IaaS, computerinfrastructure is delivered as a service. In such a case, the computingequipment is generally owned and operated by the service provider. Inthe PaaS model, software tools and underlying equipment used bydevelopers to develop software solutions may be provided as a serviceand hosted by the service provider. SaaS typically includes a serviceprovider licensing software as a service on demand. The service providermay host the software, or may deploy the software to a customer for agiven period of time. Numerous combinations of the above models arepossible and are contemplated.

Although the embodiments above have been described in considerabledetail, numerous variations and modifications will become apparent tothose skilled in the art once the above disclosure is fully appreciated.It is intended that the following claims be interpreted to embrace allsuch variations and modifications.

What is claimed is:
 1. A computing system comprising: a plurality ofstorage devices; and a data storage controller, wherein the data storagecontroller is configured to map virtual block numbers to physicallocations in a region using a fractal pattern, and wherein the regionspans two or more storage devices of the plurality of storage devices.2. The computing system as recited in claim 1, wherein the region isrepresented by a two-dimensional grid, wherein a first dimension ismeasured in blocks, and wherein a second dimension is measured instorage devices.
 3. The computing system as recited in claim 1, whereina space-filling curve is utilized to generate the block numbers in thefractal pattern.
 4. The computing system as recited in claim 3, whereinthe space-filling curve is a Hilbert space-filling curve.
 5. Thecomputing system as recited in claim 4, wherein the region is a squarewith a side length equal to 2^(k), where k is a positive integer greaterthan one.
 6. The computing system as recited in claim 4, wherein theregion is a rectangle of size A by B with M blocks, wherein A, B, and Mare integers, and wherein the data storage controller is furtherconfigured to: center the region within a square of side length equal to2^(k), wherein k is a positive integer greater than one, wherein thesquare includes N blocks, wherein N is an integer, and wherein at leastone of A or B is less than 2^(k); generate a first block number for eachof the N blocks of the square utilizing the Hilbert space-filling curve,wherein the first block numbers are numbered from 0 to N-1; discardblocks outside of the centered region; and renumber each of the M blocksof the region with a second block number, wherein the second blocknumbers are numbered from 0 to M-1 according to an ascending sequentialorder of the first block numbers of the M blocks.
 7. The computingsystem as recited in claim 6, wherein the data storage controller isfurther configured to discard the first block numbers subsequent torenumbering the M blocks of the region.
 8. The computing system asrecited in claim 6, wherein the data storage controller is furtherconfigured to generate and store a layout of the second block numbersprior to the layout being utilizing for accessing the physical locationsin the region.
 9. The computing system as recited in claim 4, whereinthe region is a rectangle of size A by B with M blocks, wherein A, B,and M are integers, and wherein the data storage controller is furtherconfigured to: partition the region into one or more square regions,wherein each square region has a side length equal to 2^(k), wherein kis a positive integer greater than one, and wherein k may vary for theone or more smaller square regions; and generate block numbers for eachof the one or more smaller square regions utilizing the Hilbertspace-filling curve.
 10. The computing system as recited in claim 9,wherein a leftover region remains after partitioning the region into oneor more square regions, wherein the leftover region is a rectangle ofsize C by D with P blocks, wherein C, D, and P are integers, and whereinthe data storage controller is further configured to: center theleftover region within a square of side length equal to 2^(j), wherein jis a positive integer greater than one, wherein the square includes Nblocks, wherein N is an integer, and wherein at least one of C or D isless than 2^(j); generate a first block number for each of the N blocksof the square utilizing the Hilbert space-filling curve, wherein thefirst block numbers are numbered from 0 to N-1; discard blocks outsideof the centered region; and renumber each of the P blocks of theleftover region with a second block number, wherein the second blocknumbers are numbered from 0 to P-1 according to an ascending sequentialorder of the first block numbers of the P blocks.
 11. A method for usein a computing system including a plurality of storage devices, themethod comprising mapping virtual block numbers to physical locations ina region using a fractal pattern, wherein the region spans two or morestorage devices of the plurality of storage devices.
 12. The method asrecited in claim 11, wherein the region is represented by atwo-dimensional grid, wherein a first dimension is measured in blocks,and wherein a second dimension is measured in storage devices.
 13. Themethod as recited in claim 11, wherein a space-filling curve is utilizedto generate the block numbers in the fractal pattern.
 14. The method asrecited in claim 13, wherein the space-filling curve is a Hilbertspace-filling curve.
 15. The method as recited in claim 14, wherein theregion is a square with a side length equal to 2^(k), and where k is apositive integer greater than one.
 16. The method as recited in claim14, wherein the region is a rectangle of size A by B with M blocks,where A, B, and M are integers, the method further comprising: centeringthe region within a square of side length equal to 2^(k), wherein k is apositive integer greater than one, wherein the square includes N blocks,where N is an integer and a least one of A or B is less than 2^(k);generating a first block number for each of the N blocks of the squareutilizing the Hilbert space-filling curve, wherein the first blocknumbers are numbered from 0 to N-1; discarding blocks outside of thecentered region; and renumbering each of the M blocks of the region witha second block number, wherein the second block numbers are numberedfrom 0 to M-1 according to an ascending sequential order of the firstblock numbers of the M blocks.
 17. The method as recited in claim 16,the method further comprising discarding the first block numberssubsequent to renumbering each of the M blocks of the region.
 18. Anon-transitory computer readable storage medium comprising programinstructions, wherein said program instructions are executable to mapvirtual block numbers to physical locations in a region of a storagesystem including a plurality of storage devices using a fractal pattern,wherein the region spans two or more of the plurality of storagedevices.
 19. The non-transitory computer readable storage medium asrecited in claim 18, wherein the region is represented by atwo-dimensional grid, wherein a first dimension is measured in blocks,and wherein a second dimension is measured in storage devices.
 20. Thenon-transitory computer readable storage medium as recited in claim 19,wherein a space-filling curve is utilized to generate the block numbersin the fractal pattern.